Let us start with some estimates. First, by \(\Delta \) -analyticity, for every \(k \geqslant 1\) , if \(G^{(k)}\)  denotes the \(k\) 'th derivative of \(G\) , then (see e.g. [1]})
\(G^{(k)}(\rho -z) \quad \mathop {\sim }_{{z \rightarrow 0\\ \mathrm {Re}(z)>0}} \quad \alpha (\alpha +1) \cdots (\alpha +k-1) \frac{c}{z^{\alpha +k}}.\) 
